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Integration. It's the simplest way on earth with which we can derive any formula like surface area or volume of symmetrical shapes and solids (square, circle, cube etc.). But what I've been hearing is that Archimedes figured out the basic formulas for area and volume in such an ancient age (many many years before Leibniz and Newton). I tried to have a small research on this topic as I was finding all this very interesting. So far I got these sources

So what I am yet thinking is that Calculus is just the pure mathematical form of observations (like dividing them all in $dx$) which Archimedes previously used in his age in non-matured form. However he succeeded in deriving those formula with these observations.

But can we really say that he applied the pure mathematical calculus in deriving all these equations? I rather think that it was his extraordinary observation which helped him.

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  • $\begingroup$ The discovery process and the rigorous proofs of Archimedes are different. The proofs are of a decidedly higher standard than those of Newton, Leibniz, Bernoulli, and their contemporaries and immediate successors. $\endgroup$ – André Nicolas Aug 6 '14 at 18:49
  • $\begingroup$ @alpha Silly me. Thanks $\endgroup$ – David H Aug 7 '14 at 14:02
  • $\begingroup$ I have read Archimedes, all the surviving works, carefully, more than once, along with a fair bit of the secondary literature. I have taught History of Mathematics quite a number of times, usually concentrating on the history of the calculus. A very large subject. $\endgroup$ – André Nicolas Aug 7 '14 at 14:04
  • $\begingroup$ It is not possible. Even an exposition of something relatively simple, such as the Quadrature of the Parabola, would take many pages. It is feasible (but not quick) to give a modern reinterpretation of the procedure, but that is too far from the original to be responsible history. $\endgroup$ – André Nicolas Aug 7 '14 at 14:15
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For fine recent works studying Archimedes and the techniques he used, see

Netz, R.; Saito, K.; Tchernetska, N. A New Reading of Method Proposition 14: Preliminary Evidence from the Archimedes Palimpsest (Part 1). SCIAMVS 2 (2001), 9-29.

Netz, R.; Saito, K.; Tchernetska, N. A New Reading of Method Proposition 14: Preliminary Evidence from the Archimedes Palimpsest (Part 2). SCIAMVS 3 (2002), 109-125.

A lot of what Archimedes does is somewhat similar to the Cavalieri principle in calculus, but Netz et al argue that he went beyond that and arguably used actual infinite sums, in a kind of a precursor of integral calculus a la Leibniz.

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